COMPUTATION FORMS 

for the class in 

Practical Astronomy 

at the Observatory of 

Columbia University, New York. 



INSTKUCTOK 
STUDENT 

CLASS OF 



COPYRIGHT 1897, BY HERMAN S. DAVIS 



&tUvWW iiiJ''^' 



57910 






\ ^ 



•jj 



rd 



Gf&Ul- 



INTRODUCTION. 



In the preparation of this pamphlet the aim has been to 
provide a skeleton merely of the formulas and forms for 
computation of such matters as are treated of in Astro- 
nomy II and the Practical Astronomy portion of the Sum- 
mer Class in Geodesy. 

The only lu'cr.-i.sai y books in addition to these Forms with 
which the student must provide himself are : 

1. The American Ephemeris and Nautical Almanac, 

2. Bremiker's (or s(niie other) Six-place Logarithm Table, 

and for collateral study one of the following 

3. Doolittle : A Treatise on Practical Astronomy, as applied to 

Geodesy and Navigation. 

4. Chauvenet: A Manual of Spherical and Practical Astronomy. 

5. Campbell: A Handbook of Practical Astronomy. 

6. Greene: An Introduction to Spherical and Practical Astronomy. 

As a book for general reference is recommended 

Young : General Astronomy, 

of the contents of which a good knowledge is supposed to 
have been obtained previous to entering the Class for \^hich 
these Forms have been prepared. 



,; 



CtfC 



Memorandum of Constants. 



Station 



LATITUDE, <1> 








h 


m 


LONGITUDE, X 






Longitude of Standard, 


>^. 




Sin <}> 




Cosec <t> 


Cos <t> 




Sec <1> 


Tan <|> 




Cotan <|> 



Transit Instrument: Thread Intervals. 

WIRES INTERVALS LOGARITHM 

f 

II 

III 

IV 
V 



Algebraical signs are applicable when 

ONE DIVISION OF THE AXIS-LEVEL 



DEFINITIONS and REMARKS. 



Aberration is the apparent change in a 
star's position caused by the progressive 
motion of light combined with the motion 
of the earth itself. 

Annual ...... is the effect produced by 

the earth's motion around the sun. 

Diurnal is the effect produced by 

the earth's motion on its axis. 

The altitude of a heavenly body is its 
angular distance above the horizon meas- 
ured on a vertical circle passing through 
that body. Single means one, sep- 
arate altitude ; though usually in sextant 
work this term is applied to the mean of 
several altitudes measured in quick suc- 
cession. Double means twice the real 

altitude, and is not used as the dual or 

plural of ' single altitude.' Equal s 

are altitudes measured when the sun is at 
equal heights above the eastern horizon 
and the western horizon. Circuinmerid- 

ian s are when the object is near the 

meridian. 

The apparent place of a star is the posi- 
tion in which the star would be seen by an 
observer at the centre of the earth. 

The azimuth ofan object is the distance 
from the north point of the horizon to the 
foot of the vertical circle passing through 
that body. A... is measured on the hori- 
zon and usually in the direction S-W-N-K. 



Chronograph. D. p. 211. 
Ca. p. 60. G. p. 32. 



Ch. II, I 71 



Colliniation or line of of a telescope 

is a straight line passing through the opti- 
cal centre of the object glass and the axis 
of rotation and _L to that axis. 

C constant in the transit instrument is 

the amount of displacement of the ' mean 
of the threads' or of ' the middle thread ' 
from the line of colliniation. 

Correction. See 'error.' 

Culmination. The transit of a heaven- 
ly body over the meridian, or highest point 
of altitude for the day. 

The declination of a heavenly body is 
its angular distance north or south of the 
equator measured on an hour circle pass- 
ing through that body. D measured 

north of the equator is + ; south is — . 8. 

Dip of the horizon is the angle of de- 
pression of the visible sea-horizon below 
the true horizon, due to the elevation of 
the eye of the observer above sea-level. 
I^et X = height of the eye in feet 

d = dip, in seconds of arc 
then d = 58^''82 \/jt: which is 

subtractive from an observed altitude. 

Eccentricity. D. pp. 180 and 196. 

Ch. II, pp. 37 and 117, Ca. p. 47. 

The ecliptic is a great circle of the cel- 
estial sphere formed by a plane passing 
through the eye of the observer (or centre 
of the earth ) and coincident with the 
plane of the earth's orbit. 



Definitions and Remarks. 



The celestial equator is a great circle of 
the celestial sphere formed by a plane 
through the eye of the observer (or centre 
of the earth) and _L to the earth's axis. 

Error vs. correction. 
A given quantity — its error = the correct, 
A given quantity -f- a correction= the correct, 
. •. Error = — Correction. 

The ho-ri'zon is a great circle of the 
celestial sphere formed by a plane through 
the eye of the observer and _L to the 
plumb-line. 

The hour angle of a heavenly body is 
the angle at the pole between the merid- 
ian and the hour circle through the body. 
This Z is measured on the equator from 
o° to 360° or from o'' to 24'' in either di- 
rection ; but -\- when toward the west. 

Hour circles ara great circles passing 
through the north and south poles of the 
celestial sphere, hence J_ to the equator. 



Latitude. 



See a later Form. 



A logarithm when representing a nega- 
tive number should have an ^ written after 
it, thus, 9.30 1030,^. Only novices use the 
negative characteristics, experienced com- 
puters use 9. 8 as given in tables of 

trigonometric functions. 

The longitude of a place on the earth is 
its angular distance from the 'zero-mer- 
idian' measured on the equator ; -|- when 
towards the west. \ or Z. 

The celestial longitude of a heavenly 
body is the arc of the ecliptic intercepted 
between the vernal equinox and the foot 
of a great circle JL to the ecliptic and pass- 
ing through the body. 



The mean place of a star at any instant 
is its position referred to the mean equa- 
tor and ecliptic of that instant. 

See ' Precession' for references. 

The meridian is that hour circle whi^h 
passes through the zenith and nadir. It 
is, therefore, J_ to the horizon as well as 
to the equator. The zero-m^eridian is 
the meridian of that place which has been 
selected arbitrarily as the origin from 
which to reckon terrestrial longitude. 

The meridian of the Observatory of 
Greenwich is that usually adopted by the 
English and American astronomers. 



Micrometer. 
Ch. II, p. 59. 



D. page 176 \ 97. 
Ca. p. 48, I 59-60. 



The nadir is the point where the plumb- 
line produced below the horizon pierces 
the celestial sphere. 

The north point is where the meridian 
intersects the northern horizon. 

Nutation. See 'Precession.' 

Obliquity of the ecliptic \s\\\^ L between 
the planes of the equator and ecliptic. It 
is about 23>^° and continually varies. 

The {diurnal) parallax of a heavenly 
body is the L at the body subtended by 
that radius of the earth which passes thro' 
the eye of the obsen^er. This Z dimin- 
ishes as the altitude of the body increases. 

Horizontal is the parallax when the 

heavenly body is seen in the horizon. 

Polar distance is the complement of the 
declination. p = 90° — 8. 

Precession. D. p. 558, chapter x. 
Ch. I, chapter xi. Ca. chapter v. 



Definitions and Remarks. 



Prime vertical. See 'Vertical circle.' 



Proper motion. 
Ch. I, p. 620. 



D. page 57S, ^. 334- 
Ca. page 38. 



The rate of a chronometer is the (daily 
or) hourly change in the chronometer- 
correction. Hence, if the chronometer is 
gaining, the rate is — ; and if losing, +. 

Refraction. The usual formula is : 
r= J/X W X T^ X T^ 

the primes being to distinguish these letters from the tabular ones 

lait these factors of J/diflfer so little from 
unity that in the Table we have put : 
B = B^ - \\ t=:t'-i; T=T—\ 
By substitution of these we get, quite 
accurate enough for sextant work and far 
easier to calculate mentally, 

r= J/-f- J/X IB \-T \- T\ 

Th3 right ascension of a heavenly body 
is the arc of the equator intercepted be- 
tween the vernal equinox and the foot of 
the hour circle through that body. It is 
reckoned always eastwardly from the ver- 
nal equinox, from o'^ to 2^. 



Sextant. D. page i8_; 
Ch. II, chapter iv. 



Ca. page 6i- 
G. page 42. 



Time. 



See later Form. 



Transit instruvient. D. page 267. 

Ch. II, chap. V. Ca. chap. viii. G. p. 23. 

The vernal equinox is that point of in- 
tersection of the equator and the ecliptic 
through which the sun appears to pass in 
the spring ( about March 20"" ) in going 
from the south to the north of the equator. 



Vernier. 



D. p. 174. Ch. II, p. 



A vertical circle is a great circle passing 
through the zenith and _L to the horizon. 
The prime vertical is that vertical circle 
which passes through the east and west 
points. 

The zenith is the point where the plumb 
line produced pierces the celestial sphere 
above the horizon. The zenith distance 
of an object is its angular distance from 
the zenith measured on the vertical circle 
passing through the object. X, = 90°— h 



D. = Doolittle 



ABBREVIATIONS 

Ch. := Chauvetiet Ca. = 



Campbell G. = Greene. 



K 



Fundamental Formulas of the Z-P-S Triangle. 



Given the spherical triangle whose vertices are at the 
zenith, the pole, and the star (or sun). This is known as 
the Z-P-S-triangle. 

Let the side ZP = 90° — c^ 

PS = 90° — 6 

S2 = 90° — h 
/ p = t Hour angle 
^ S = q Parallactic angle 
L 1 = 1 

By the theorem : The sines of the sides of a spherical 
triangle are proportional to the sines of the angles opposite ; 
we have : 

cos h sin Z = sin t cos 8 (1) 

cos h sin S = sin t cos <j) (2) 

cos 8 sin S = cos ^ sin Z (3) 

By drawing a great circle through S perpendicular to 
PZ at the point K, and applying Napier's Rule first to the 
A SZK and then to the A SPK, we find : 

sin h = sin 8 sin <j) -|- cos 8 cos <() cos t (4) 

sin 8 = sin <}> sin h -\- cos <j> cos h cos Z (5) 

sin <j> = sin h sin 8 -|- cos h cos 8 cos S (6) 

Draw a great circle through S and the ' north point ' de- 
signated as R, then will ZR = 90° and from the A SPR 
we find : 

cos SR = sin 6 cos </> — sin ^ cos 5 cos t 
Likewise, in the A SZR : 

cos SR = cos h cos Z. 
Equate these two values and there results : 

cos h cos Z = sin 8 cos <|) — sin 4) cos 8 cos t (7) 

and by analogy, 

cos 8 cos S = sin <|> cos h — cos <}) sin h cos Z (8) 

cos <j> cos t = sin h cos 8 — cos h sin 8 cos S (9) 



LATITUDE. 



I. The astronomical latitude of a point on the earth's sur- 
face is the angle formed with the plane of the equator by a 
plumb-line at the given point. 

II. Celestial latitude is the distance of a heavenly body 
north or south of the ecliptic measured on a great circle per- 
pendicular to the ecliptic. Its symbol is p. 

III. The geocentric latitude of a point on the earth's surface 
is the angle formed with the plane of the equator by a line join- 
ing the point with the earth's centre. Its symbol is <|>^. 

IV. The geodetic or geographical latitude of a point on the 
earth's surface is the angle made with the plane of the equator 
b}' a normal to the surface at this point. Its symbol is <j>. 

The geodetic latitude is always greater than the geocentric; 
the difference, <j) — 4>^, is the ' reduction of the latitude.' 

The astronomical latitude and the geographical are usually 
assumed to be eqiial ; localities where this assumption is not 
true are exceptional, but do exist. 

The latitude determined by the methods of these Forms is 
the geodetic. 



Latitude by "Single Altitudes" of the Sun. 



(4) 



Given the fundamental equation, 

sin h = sin ?> sin </> -4- cos 5 cos cos t 
to solve for the value of </>. 

Assume the quantities d and D such that : 

d sin D = sin 5 

d cos /) = cos 6 cos if 

y = <\, — D 

By substitution in the fundamental equation, we have : 

cos Y = sin h cos 5 sin D. 
Dividing the first assumed equation by the second, we get : 

tan D = tan 5 sec t 
From these two derived equations are obtained the numerical values of v and D 
and, therefore, 

<|) = D ± 7. 



REFERENCES 

Chauvenet, Vol. I, Page 310, \ 206, §164. 
Doolittle, Page 236, ^ 243. 
Campbell, Page 77, | 89. 
Greene, Page 82, I 119-121. 
Young, ^ 103. 



I. ) Z? is to be always taken <90°, 
its algebraic sign being deter- 
mined by the sign of tan D. 

2. ) As 7 is determined by its cos. 
it may be either -|- or — and 
that sign must be used which 
gives the most reasonable <l>. 

3. ) The sign of t is immaterial as 
always t <^6'^. 



r 



^ 



Latitude by "Single Altitudes" of the Sun. 



Watch Comparisons. 
Chron Watch .... 



BEFORE OBS 

CHRON, 

h m s 


SERVING. 

WATCH 

HI S 


AFTER OBS 

CHRON. 


ERVING. 

WATCH 










































1 






Watch ^"^^ 

SLOW 


m s 


Watch '^*^-^ 

SLOW 


m » 



Hourly rate of watch = 

Watch correction at ^ 

time of observation J 



Index 


Correction. 


ON ARC 

/ // 




OFF ARC 

/ // 


— 




4- 


— 




4- 


— 




-h 


— 




4- 


— 




-f- 


3 

i 




4- 



Yz Algebraic 
sum of means 



Barometer 
Attached Therm. 
External Therm. 



inches. 



D ate Station Sextant No. 

Observer Recorder Object 

DOUBLE ALTITUDE TU^^ 



Latitude by "Single Altitudes" of the Sun. 

ooiyn:PTJT.i^Tionsrs. 



Obs. Doub. Altitude 




Observed Watch time 


Index Correction 




Watch Correction 


Eccentricity 




Clock Correction 


Corr'd Doub. Alt. 


Red" from Stand, to Mean 


Observed Alt., h' 




Mean time of obs. 


Refraction, r 




Equation of time 


Semi-diameter, s 






Parallax in Altitude 




Apparent time of obs. 


Corrected Alt, // 




Hour angle, t (time) 

make t plus if sun west of meridiau 

t (in arc) 








Decl. Gr. Mn. noon 




Hourly- change ( 


I 


sec / 


Inten-al of time ( 


i 


tan h 


0'^ 6 at obser\'ation 


tan D 




)n 




Eq. of Time, Gr. Mn. noc 




Hourly change ( 


1 


cosec « 


Interval of time { 


1 


sin D 


Eq. of Time at observation t 


sin li 


Mean Refraction, J/ = 

1 




cos V 


Factor, B j 






1 


71/ = 


/ D (ever < 90°) 


„ . i 




Z y 


Sum, r 




Latitude, Z> ± y 



Latitude by "Single Altitudes" of the Sun. 

nsroTES. 



I 



Latitude by "Circummeridian" Altitudes of the Sun. 



(4) 



From the fundamental equation, 

sin h = sin 6 sin </> -I- cos 5 cos <A cos / 
we get sin h^ — sin h ^= 2 cos 4> cos 5 sin^ % i 

by substituting cos t = i — 2 sin""^ % t and iif -^ ± 5 -^ 93° — h^ 

where h^ is the meridian altitude, i. e. the altitude when / = o ; and where h is the 
observed altitude at hour angle t. For brevity write sin h = sin h^ — y in 
which it is seen that h r=f[y). Remember that h-^ is a constant on any par- 
ticular day of observation. By Maclaurin's theorem, and after substitution of the 

values of dh/dy, d^h/dy\ obtained by differentiating sin h = sin h^ — y 

we have h ^= h^ — y sec h^ -{- % y'^ tan h^ sec^ h^ 4" 

Restoring the value of y and substituting the value of (sin h-^ — sin h ) we get 
h^ = //-{- cos </) cos 5 sec h^ X 2 sin''^ Yz t cosec V 

— (cos cos 5 sec /^i)^ X 2 sin* >^ t cossc i^^ X tan /z^ -f 

For brevity this may be written h^ = h A^ Qpt — D« -}- where in 

and ?/ are given in the Tables. Since the mean of several measured altitudes 
may be used, this becomes h^=^ h -\- Qni^ — DiIq -\- where 

mQ= \_m^\-m.^-\-... wj/r and n^^= \_n^-{-n.2 -{-... n^']/r. r = no. of observations. 

Then 4, = 8 -h 90° - [h -f (Cwq - D;/^)] 



Watch Comparisons. 
Chron Watch .... 



BEFORE OBS 

CHRON. 


SERVING. 

WATCH 


AFTER OBSI 

CHRON. 


:rving. 

WATCH 




































- 






C 






Watch -; 


m s 


Watch ""^^ 

SLOW 


m s 



Hourly rate of watch = 

Hence, assumed watch error at T^. = 

• gains 

From first comp. to T,,/ watch 10^^^ 

At T,^/ accurate ' watch error' is 

Error of assumed 'watch error' = 



Index Correction. 



ON ARC 

/ // 


OFF ARC 

/ // 


— 


+ 


- 


+ 


- 


+ 


- 


+ 


- 


4- 


i ^ 


+ 


Yz Algebraic ) 




sum of means j 





Barometer inches. 

Attached Therm. ° 

BxTERNAi. Therm. ° 



Latitude by "Circummeridian" Altitudes of the Sun. 


A" is a factor introduced to take account 
of the rate of the watch while observing. 
Chauveriet, Vol. I p. 241. Doolittle, \ 152. 


Computation of C and D requires the use 
of an approximate latitude. This may be 
obtained by reducing the highest observ- 
ed altitude as if it were a meridian altitude. 


REFERENCES 

Chauvenet, Vol. I, Pages 233-53, ^ 167-75. 
Doolittle, Pages 238-255, \ 150-152. 
Campbell, Page 79, | 91. 
Greene, Page 84, \ 122-124. 


Chauvenet, Vol. I ^ 161. Doolittle, f 141. 


Max. double altitude 
Index Corr. + Eccen. 


Corrected doub. alt. 


Max. alt. observed 
r -\- s -{- Parallax 


Assumed merid. alt. 


Assumed zenith dist. 
Declination, 6 


Approx. latitude, ^' 


Date . . Station vSkxtant No 


Observer RE 

DOUBLE ALTITUDE 


CORDER Objec 

TIME 


X 

^ observed -*- w 












































L. 









r 



Latitude by "Circummeridian" Altitudes of the Sun. 



Eq. of Time, Gr. Aj 
Hourly change ( 
Longitude ( 
Eq. of time local a; 


)p. noon 

1 


Mean Refraction, iJ/ = 
Factor, B | 


3p. n.,E 


1 

„ T J 


O'^Culmin. app.time /2 OO OO.O 


Sum, r 


Equation of time, ] 
Chron. error (ix^ mear 

0'* culm, by chror 
Approx. watch em 
Approx. culm., T,,. 
Error of 'approx. w 
Culm, by watch, T, 


I time 


Decl, Gr. App. noon 
Hourly change ( \ 
Interval of time ( j 


i.,T, 
or 


0'^ 5 at /fzw^ of obs. 


atch error' 




Obs. Doub. Altitude 




Index Correction 
Eccentricity 




n 


Log. K 
Cos./.^ 

Cos 5 

Sec/z 


ni 


Corr'd Doub. Alt. 










Observed Alt., h^ 
Refraction, r 




Log. C 
Log. mo 
Log. Cwo 
C^o = + 


Semi-diameter, s 
Parallax in Altitude 




Corrected Alt., h 
(C^o - Tin,) 




Log. C^ 

Tan/-! 

Log. ;^o 




Reduced altitude h^ 




C = go° - /i. 






i 


Log. D«o 
D;/o = 4- 


8 


s 




Latitude, <}> 



Latitude by ''Circum meridian'' Altitudes of the Sun. 



TIME. 



I. A sidereal day is the interval between two successive 
transits of the vernal equinox over the upper branch of the mer- 
idian. This day begins at differing intervals from noon accord- 
ing to the time of the year. 

The sidereal time at any meridian is the hour angle of the 
vernal equinox at that meridian. It is counted from o'' to 24*^. 

II. An apparent solar day is the interval between two suc- 
cessive transits of the real sun over the upper branch of the 
meridian ; this day counts from midnight. 

The appaient time at any meridian is the hour angle of the 
real sun at that meridian, and is counted twice from o'^ to 12^ ; 
the first series being A. M., and the second series P. M. 

III. A mean solar day is the interval between two successive 
transits of the 'second fictitious sun,' i.e. the 'mean sun,' over 
the upper branch of the meridian. This day begins at midnight. 

The fnean time at any meridian is the hour angle of the 
* mean sun ' at that meridian. 

The equation 0/ time is the quantity which must be added 
algebraically to the apparent time to produce the mean time, 

Standard time is the same as mean time, except that it is 
constant for each lune of 15° in longitude reckoning from the 
meridian of Greenwich. 

IV. The civil day is reckoned from midnight to midnight. 
V. The astronomical day is from noon to noon and is 12'' 

behind the civil day, i. e. it runs from noon of the same civil day 
to noon of the following civil day. 



Conversion of Time. 



A. 


B. 


standard to Sidereal. 


Sidereal to Standard. 


Station 


Station 


Civil. Date 


AsTRON. Date 


h m s 


h m s 


Standard time, T^ 




{K - M 




Local mean solar, T 


Sidereal time, 


Sid. time of m. n. at) 
the zero-meridian V^ ] 


V. 


4-9^8565 A (Table III) 


4-9'8565 ^ (Table III) 


Sid. time of m. n. ) 
at the Station, V \ 


V 


Interval from noon, 

T IF P.M. T - 12 IF A.M. 


® - V 


+9^8565 X Int. (Table III) 


-9«8296(©- F) [Table II] 


Sidereal time, 


Int. from local m. n. 


Astronomical Date= 


Mean time, T 


[Civil Date — i^ if r — 12 is used,] 
[Civil Date \i T is used.] 


^-As 




Standard time, 7; 




Civil Date = 




[Astron. Date -|- i"^ if later than noon,] 
[Astron. Date if earlier than noon.] 



REFERENCES 



Ckauvenet, Vol. I, Pages 52-64, ^ 37-53. 
Doolittle, Pages 162-173, 'i 89-95. 
Campdelly Pages 6-10, ^ 4-12. 
Greene, Pages 13-20, ^ 39-44, ^92. 



Conversion of Time. 



Standard to Apparent. 

Station 
Civil. Date 



Standard time J", 

Local mean time, T 
Long, from Green., a 



time elapsed since greenwich meau noon 

T (in hours) 



Eq. of time, Hourly change, aE 



Eq. of time, Gr. mean noon 
aE X T^g = Change in Eq. = 
Required Eq. of time, E 
Local mean time, T 



Apparent time, T—% 

Civil Date 



Apparent to Standard. 

Station 
Civil. Date 



Apparent time, T 



interval since greenwich apparent noon 

Tg (in hours) 



aE 



Eq. of time, Gr. app. noon 
aE X Tg = Change in Eq. 

E 
Local apparent time, T 
Mean time, T + E 



Standard time, 7"^ 

Civil Date 



A = Longitude from the zero-meridian, usually that of Greenwich, -j- if -^^st. 
Table II and Table III are given at the end of the Atnerica^i Ephem,eris. 



^ 



Time by "Single Altitudes" of the Sun, 

:FOK.lN^TJILi^S. 



From the fundamental equation : 

sin h = sin 6 sin «^ -t- cos 6 cos <f) cos t (4) 

we have 

, sin h — sin 6 sin*;*/* 

cos t = ^ 

cos 5 cos «/> 

Subtract each member from unity, and substitute i — cos i=: 2 sin'' }4 t, 



Then 



2 sin'^ K ^ = cos ( «/> — 5 ) — sin ^ ^ c os ( (^ — 6 ) — cos g 

cos 6 COS <^ cos 6 COS ^ 



Since, in general, cos a — cos b ■= 2 sin ^ ( a h- ^ ) sin >^ {a — b) 

Olnefs Trig. Page 30, g 59 
we have 



sm>^t= / sin K [ t -H ( <!> - S )] sin K [ t - ( <!> - 8 )] 
>J cos S cos ()> 

REFERENCES 

Chauvenet, Vol. I, Page 209, | 146 
Doolittle, Page 216, ^ 122-124 
Greene., Page 9, ^ 34 
Young, ^116 
Campbell, Page 75, ^ 86. 



r 



Time-by "Single Altitudes" of the Sun. 





Watch Comparisons. 




Chron 


Watch 




BEFORE OBSERVING. 


AFTER OBSERVING. j 


CHRON, WATCH 

h m s 1 m 8 


CHRON. 

h m s 


WATCH 

su s 






i.; h- '.-' ■ 








.................... 





m s 


{ ? 


m s 


Watch ;- 


Watch -^^"^ , 

■ - SLOW- ^ 




Hourly rate of watch = 






Watch correction at ) 

time of observation / 







Index Correction. 



OFF ARC 

/ // 



^ 



V2 Algebraic ) 
sum of means / 



Barometkr 
ATTACikfei) Therm. 
BxTKRNAi, Therm. 



inches. 



Date Station SkxtanT No. 

Observer Recorder Object 



DOUBLE ALTITUDE 



TIME 



Time-by **^Sirtglc Altitudes" of the Sun. 

oo:]vi:F^TfT.^'i'ionsrs_ 



/ // 

Obs. Doub. Altitude 

Index Correction 

Eccentricity 


Mean Refraction, 7J/=^ 
Factor, B \ 


Corr'd Doub. Alt. 


Observed Alt., h' 


Sum, r 


Refraction, r 
Semi-diameter, 5 
Parallax in Altitude 


Decl. Gr. Mn. noon 
Hourly change ( \ 
Interval of time ( j 






Corrected Alt., h 
I^atitude, ^ 


0'^ ^ fit observation 


Declination, 5 


Eq. of Time, Gr. App, noon 


90° - /z = < 

^ — 5 


Hourly change ( | 
Interval of time ( ) 


<+ (*-S) 


Eq. of Time at observation t 


^-(<f.-5) 


± >< t (in arc) 

± t (in arc) 

Hour angle, t (time) 


sin >4 [^4- (</.— 5)] 


make t plus if suu west of meridian 


sin >2 [^ — (<^ ~ 0] 
sec </) 


Apparent time of obs. 
Equation of time 


sec 5 


Mean time of obs. 


sin2 y, t = 


Chronometer time 


sin y,t 


Chron. correction 



Time by "Equal Altitudes" of the Sun. 



Let 6 = the Q'* declination at apparent (local) noon, 

AS = the hourly change in Q"" ^ from meridian to time of observation, 

positive if sun is nioviag uorthward ucgalivu if soulhwurd 

i = the elapsed time between a. m. and p. m. observations, 
Tq = mean of the morning and afternoon chronometer times, 
aT(, = a-\- d = the change produced in Tq by the change in 0'* 5. 
Then ^ l-\- aTq = hour angle (reckoned toward the easl) at a. m. observations 
^2 t — A 7"o =^hour jingle (reckoned toward the wesl) at p. m, observations 

5 — }i /fA6 = sun's declination at morning observation, 

6 -{- }^ IaS= sun's declination at afternoon observation. 
Substitute these quantities in the fundamental equation 

sin k = sin s sin <t> -I- cos 5 cos 4t cos / (4) 

for morning and afternoon observations separately, and we have 
sin /z = sin <f.sin(6 If >^ /f A6) -f cos<^ cos(6 qi i^ / AS) cos(>^ /* ± aT;) for ""'^"i'^ft^.ioou 
Expand each of these expressions; subtract the upper from the lower; transpose 
and divide by the coefficient of sin aTq , remembering that usually we may put : 
i^ / A6 = tan >^ ^ A6 A 7; = sin A 7; i = cos a Tq 

We get finally : 

ATo = IT — ^^^*-, A8 tan (|» + ___/^A^ AS tan 8 "°^" 

" ^ 15 sin >^ t 15 tan >^ t midnight 

The logarithms of these fraction-coefficients, designated as A and B, are griven in 

the Tabi.es. 

REFERENCES 

Chauvenet, Vol. I, Page 198, \ 140-144 
Doolittle, Page 230, ^ 137-139 
Campbell, Page 71, ^ 82-84 
Greene, Page 58, 'i 94-95 



Great care must be taken that the morning and afternoon observations be made on 
the same limb of the sun. 



Time by "Equal Altitudes" of the Sun. 



Watch Comparisons. 
Chron Watch ... 



BEFORE OB! 

CHRON. 


SERVING. 

WATCH 


AFTER OBSERVING. 

CHRON. WATCH 

h m s ui, s 
































_ . _ 






1 






Watch -^^^^ 

SLOW 


lU s 


Watch -^^^-^ 

SLOW 


m s 



Hourly rate of watch = 

Watch correction at 1 

time of observation J 



Index Correction. 





ONARC^^ 


OFF ARC 
/ // 


— 




+ 


— 




+ 


— 




-h 


— 




4- 


— 




4- 


c 


+ 


}4 Algebraic 1 
sum of means J 



Barometer 
Attached Therm. 
BxTERNAi. Therm. 



inches. 



Chron 


Watch Comparisons. 

WaTpw 




Index Cc 

ON ARC 

/ // 


>rrection. 

OFF ARC 


BEFORE OB; 

CHRON, 

h 111 s 


SERVING. 

WATCH 


AFTER OBS 

CHRON. 


ERVING. 

WATCH 


/ // 

4- 
4- 










4- 










4- 










4- 










1 — 


4- 












1 

a 






)4 Algebraic \ 
sum of means ( 


. s 


» • 




Watch "^^^"^ 

SLOW 

Hourly rate c 

Watch correc 
time of obsei 


f watch = 

tion at ) 

•vation / 


Watch -'„ 




Barometer inches. 
ATTACHED Therm. ° 
Extern A I. Therm. ° 



r 



Time by "Equal Altitudes" of the Sun, 

OBSJBi^"V"^Tioj:Nrs- 



Date , Station 



Sextant No. 



Observer Recorder 

TIME DOUBLE ALTITUDE 



Object ... 

TIME 



OOl^IPXJT^TIOISrS. 



Eq. of Time, Gr. App. noon 
Hourly change ( | 


A. M. Watch time 
Watch correction 


Inter\^al of time ( j 


A. M. Chron. time 


Eq. of Time at local „;iX".ht 
App. time of App. ^,^Z%ut 
Mean time of local App. 

uoou or niiduight 


P. M. Watch time 
Watch correction 


P, M, Chron. time 


Sum of Chron. times 
Diff. is Elapsed time, t 
y^ Sum = 7^3 


Decl. Gr. App. noon 


Honrh' change ( ^ 
Interv'aloftime ( ] 


Log A Log B 
Log AS Log AS 
Tan ^ Tan 5 


G" ^ at noon or midn't 


Log a Log b 


Daih' change in H. Diff. in S 
1/24 diUo = Vs 


Middle Chron. time, T^ 






Long, or (Long -f- 12'^) X a^s 
H. Diff. for preceding noon 


Sum = Chron. time 
Mean time of App. n^Xikht 






Sum = AS y^l^^S^S^^ 


Chron. correction 



^ 

Time by ''Equal Altitudes" of the Sun. 

nsroTES. 



From the fundamental equation : 

sin // = sin 5 sin <l> -h cos 6 cos <^ cos f (4) 

we have 



cos s cos ^ 
Subtract each member from unity, and substitute i — cos 1=2 sin^ )4 A 
Then 

2 sin^ }^ t = cos ( .A — O — sin A ^ c os ( — 6 ) — cos < 

cos 8 cos <A COS 5 COS <A 

Since, in general, cos a — cos b = 2 sin j^ ( « h- <5' ) sin % [a — b ) 

Olney's Trig. Page 30, §59 
we have 

sinKt= / sin K [ t + ( <t > - 8 )] sin K [ t - ( <{> - 8 )] 
>J cos S cos <{> 

REFERENCES 

Chauvenet, Vol. I, Page 206, ^ 145 
Doolittle, Page 220, | 125 
Greene, Page 9, ^ 34 
Campbell, Page 74, | 85. 



■\ 



Time by "Single Altitudes'* of a Star. 



Watch Comparisons. 
Chron Watch .... 



BEFORE OB; 

CHRON, 

h m s 


SERVING. 

WATGH 

m s 


AFTER OBS 

CHRON. 


ERVING. 

WATCH 




































" 






i 

a 






Watch ""^^ 

SLOW 


m s 


watch ;••;, 


m a 



Hourly rate of watch = 

Watch correction at 
time of observation 



index Correction. 



4- 



}4 Algebraic 1 
sum of means J 



Barometer 
Attached Therm. 
BxTERNAi. Therm. 



inches. 



Date Station Sextant No. 

Observer Recorder Object 

DOUBLE ALTrXUDE TIME 



V. 



Time by "Single Altitudes" of a Star. 

oonvniPXJT^^Tionsrs. 



Obs. Doub. Altitude 

Index Correction 

Eccentricity 


Mean Refraction, 7)/ = 
Factor, B ] 

. - i 

Sum, r 




Corr'd Doub. Alt. 




Observed Alt., h^ 




Refraction, r 


^'* Right Ascension '^ 

t'pbenieris part ii 

i^'' Declination ° ^ 




Corrected Alt, // 

Latitude, <^ 

Declination, 5 




± >< z* (in arc) 
± / (in arc) 




Hour angle, t (time) 

make t plus if stitr wet t of meridian 

"2^' '^ Right Ascension 




90°-//=^ 
^+ (cf.-S) 




Sid. time of obs., 




SEE FORM 'sidereal TO MEAN TIME' 

Sid. time Gr. m. n., V^^ 
Corr" for Long. (Table III) 


%ii- u-^)^ 


Sid. time local va.. n., V 


sin >^[^--f (0- 5)] 
sin Yzli — (</> — 5)] 


Sid. Interval, — V= 
Red'^ to mean time, ( Table II ) 


sec ^ 


Mean time of obs., 7",„ 


sec S 


Standard time, T, 
Chronometer time 




sin'-^ y.t^ 




sin ^ / 


Chron. correction 





Time by ** Single Altitudes" of a Star. 

ISrOTIES. 



T^^BXjE 1 



BesseCs Correction for Refraction. 



Bessel's Mean Refraction : Barom. = 29.6'°; Ther. = 48.75'^ 



Correction-Factors for 



Alt. 


Refr. 


Alt. 


Ccfr. 


Alt. 


R2fr. 


Alt. 


Refr. 


Barometer 


Thermometer 


h' 


M 


h' 


iV 


h' 


M 


h' 


AT 1 INCHES 


B 


FAKH 


T 


20° 0' 


15^3 


26^ 0^ 


1 1 Y^o 


36^ 0^ 


79''3 


48° 0^ 


5i'^9 


28.0 


- .054 


20° 


-f -060 


10 


155-9 


10 


117.0 


23 


78.4 


49 


50.2 


.1 


-.051 


22 


+ -055 


20 


154-5 


20 


116.1 


43 


77-4 


50 


48.4 


.2 


~ -047 


24 


-j- .051 


30 


153-2 


30 


II5-3 


37 


76.5 


51 


46.7 


•3 


— -044 


26 


4- -047 


40 


151. 8 


40 


114-5 


23 


75-6 


52 


45-1 


-4 


— .040 


28 


4- .042 


53 


150.5 


50 


113. 6 


43 


74-7 


53 


43-5 


•5 
.6 


-.037 
-■034 


33 


+ .038 


21 


149.2 


27 


112.8 


38 


73.8 


54 


41.9 


•7 


— .030 


32 


+ .034 


(O 


148.0 


10 


112. 


23 


72.9 


55 


40.4 


.8 


-.027 


34 


4- .030 


20 


146.7 


20 


111.2 


43 


72.0 


56 


38.9 


.9 


-.024 


35 


-J- .026 


30 


145-5 


33 


110.5 


39 


71.2 


57 


37-5 


29.0 


— .020 


33 


"t- .022 


40 


144-3 


40 


109.7 


23 


70.3 


580 


36.1 


.f 


-.017 






53 


143- - 


50 


108.9 


43 


69-5 


59 


34-7 


.2 
.3 

.4 


— .013 


40 


4- .017 
+ .013 


















— .010 


42 


22 


141. 9 


28 


108.2 


43 


68.7 


63 


33-3 


-.007 


44 


4- .009 


10 


140.7 


20 


106.7 


20 


67-9 


6i 


32.0 


•5 


-.CO3 


46 


4-.C05 


20 


139-6 


40 


105.2 


43 


67.1 


62 


30.7 


.6 


ip .COO 


48 


4- .002 


3^ 


138.5 


29 


103.8 


41 


66.3 


63 


29.4! 


•7 


-\- .003 






40 


137-4 


23 


102.4 


20 


65.6 


64 


28.2 


.8 


H- .007 


53 


— .002 


50 


136.2 


40 


lOI.O 


43 


64.8 


65 


26.9 


.9 
30.0 


-l- .010 
4- .014 


52 
54 


-.006 
— .010 


23 


135-2 


30 


99-7 


42 


64.0 


66 


25-7 


.1 


-r -017 


56 


-.014 


10 


134- 1 


20 


98.4 


23 


63-3 


67 


24-5 


.2 


4- .020 


58 


— .018 


20 


133-0 


43 


97.1 


40 


62.6 


68 


23-3 


•3 


T--024 






30 


132.0 


31 


95-8 


43 


61.8 


69 


22.2 


.4 


~r -027 


60 


— .022 


40 


131. 


20 


94-6 


20 


61. 1 


70 


21.0 


•5 


-h -031 


62 


-.025 


50 


129.9 


40 


93-3 


40 


60.4 


71 


19.9 


.6 


+ -034 


64 


— .029 
















•7 


-f-037 


65 


--033 


24 


128.9 


32 


92.1 


44 


59-7 


72 


18.8 


.8 


+ .041 


63 


— .036 


(O 


127.9 


23 


91.0 


23 


59-0 


73 


17-7 


-9 


+ .044 






23 


127.0 


43 


89.8 


40 


58.4 


74 


16.6 


31.0 


+ •047- 


70 


— .040 


30 


126.0 


33 


88. 7 


45 


57-7 


75 


15-5 




72 


-.044 


40 


125.0 


20 


87.6 


20 


57-0 


76 


14-4 . 




74 


— .047 


50 


1 24. 1 


40 


86.5 


40 


56.4 


77 


JO-, Attach 


ed Therm 


76 


-.051 




123.2 


34 


85-4 


46 


55-7 


78 


12.3 


FAHR 


T 


78 


— -054 


25 








10 


122.3 


23 


84.3 


20 


55-1 


79 


II. 2 


30° 


± .oco 


8c 


-.058 


20 


121. 3 


40 


83-3 


40 


54-4 


80 


10.2 


40 


— .001 


82 


— .061 


30 


120.4 


35 


82.3 


47 


53-8 


81 


9-1 


50 


— .002 


84 


— .065 


40 


119.6 


20 


81.3 


20 


53-2 


82 


8.1 


60 


-.C03 


86 


— .06S 


50 


118.7 


40 


80.3 


40 


52.6 


83 


7-1 


70 


-.003 


88 


-.071 
















80 


-.004 






26 


117.S 


36 


79-3 


48 


51-9 


840 


6.1 


1 90 


-.005 


90 


-•075 



Refraction = .r==^'J/4-iJ/(^4-T 4- T) H 
Within the limits of this table this last term is <; o, "]' 



3T\_BT-\-r{BT-{- T^B)-]. 

and may be neglected in sextant work. 



Tj^BLE 2 



Values of Log A and Log B for "Equal Altitudes.' 



F 
A- 


or noon, 
- B-h 


ARGUMENT = 


= ELAPSKD TIME. 


For midnight, 
A+ B + 


m 


3^ 


4h 


5^ 


6h 


7^^ 


8h 




Log A 


IvOgB 


Log A 


LogB 


Log A 


LogB 


Log A 


LotrB 


Log A 


LogB 


Log A 


LogB 


o 


9.4172 


9.3828 


9.4260 


9-3635 


9-4374 


9-3369 


9-4515 


9.3010 


9.46S5 


9- 253c 


9.4884 


9.1874 


2 


.4174 


.3822 


.4263 


•3627 


•4378 


•3358 


.4521 


.2996 


.4691 


.2511 


.4892 


.1848 


4 


.4177 


•3«i7 


.4266 


.3620 


•4383 


•3348 


•4526 


.2982 


.4697 


-2492 


.4899 


.1822 


6 


•4179 


.3811 


.4270 


.3612 


•4387 


.3337 


•4531 


.2968 


.4704 


•2473 


.4906 


.1796 


8 


.4182 


.3806 


.4273 


.3604 


•4391 


.3327 


•4536 


•2954 


.4710 


.2454 


•4913 


.1769 


iO 


9.4184 


9.3800 


9-4277 


9-3596 


9-4396 


9-3316 


9-4542 


9.2940 


9.4716 


9^2434 


9.4921 


9.1742 


12 


.4187 


•3794 


.4280 


•3588 


.4400 


.3305 


.4547 


-2925 


•4723 


.2415 


.4928 


•1715 


•4 


.4190 


.3789 


.4284 


• 3580 


•4405 


.3294 


•4552 


.2911 


.4729 


•2395 


-4935 


.1687 


i6 


•4193 


.37H3 


.4288 


.3572 


.4409 


-3283 


.4558 


.2896 


•4735 


•2375 


-4943 


.1659 


i8 


.4195 


•3777 


.4291 


•3564 


.4414 


.3272 


.4563 


.2881 


-4742 


.2355 


-4950 


.1630 


20 


9.4198 


9-3771 


9-4295 


9-3555 


9.4418 


9.3261 


9^4569 


9.2866 


9-4748 


9-2334 


9-4958 


9.1602 


22 


.4201 


.3765 


-4299 


.3547 


•4423 


-3249 


•4574 


.2850 


•4755 


-2313 


.4965 


•1573 


24 


.4204 


•3759 


.4302 


-353H 


.4427 


.3238 


• 4580 


.2835 


-4761 


.2292 


•4973 


.1543 


26 


.4207 


.3752 


.4306 


.3530 


.4432 


-3226 


•4585 


.2819 


-4768 


.2271 


.4980 


•1513 


28 


.4209 


.3746 


.4310 


.3521 


•4437 


-3214 


•4591 


.2804 


-4774 


.2250 


.4988 


-1483 


30 


9.4212 


9- 3740 


9-4314 


9-3512 


9.4441 


9.3203 


9^4597 


9.2788 


9.4781 


9.2228 


9.4996 


9-1453 


32 


.4215 


•3733 


•4317 


-3503 


•4446 


•3191 


.4602 


.2772 


-4788 


.2206 


-5003 


.1422 


34 


.4218 


•3727 


.4321 


-3494 


•4451 


-3178 


.4608 


.2756 


.4794 


.2184 


.5011 


.1390 


36 


.4221 


.3720 


-4325 


-3485 


•4456 


.3166 


.4614 


•2739 


.4801 


.2162 


•5019 


•1359 


38 


.4224 


-3713 


.4329 


-3476 


.4460 


•3154 


.4620 


2723 


.4808 


.2140 


.5027 


.1327 


40 


9.4227 


9-3707 


9-4333 


9-3467 


9-4465 


9-3142 


9.4625 


9.2706 


9-4815 


9.2117 


9-5035 


9.1294 


42 


.4231 


.3700 


-4337 


•3457 


.4470 


.3129 


.4631 


.2689 


.4821 


.2094 


-5042 


.1261 


44 


•4234 


-3^93 


•4341 


-3448 


-4475 


.3116 


•4637 


.2672 


.4828 


.2070 


•5050 


.1228 


46 


.4237 


.3686 


-4345 


• 3438 


.4480 


-3103 


•4643 


-2655 


• 4835 


.2047 


•5058 


.1194 


48 


.4240 


-3679 


-4349 


•3429 


-4485 


.3091 


•4649 


-2638 


.4842 


.2023 


.5066 


•I 159 


50 


9-4243 


9.3672 


9-4353 


9^3419 


9.4490 


9.3078 


9-4655 


9.2620 


9.4849 


9.1999 


9-5074 


9. 1 125 


52 


.4246 


-3665 


-4357 


•3409 


-4494 


.3064 


.4661 


.2602 


.4856 


-1974 


.5082 


.1089 


54 


.4250 


•3657 


.4361 


•3399 


-4500 


-3051 


.4667 


.2584 


.4863 


•1950 


•5091 


.1054 


56 


.4253 


.3650 


.4366 


.3389 


-4505 


•3038 


•4673 


.2566 


.4870 


.1925 


-5099 


.1017 


58 


.4256 


•3643 


-4370 


•3379 


.4510 


.3024 


.4679 


.2548 


•4877 


.1900 


•5107 


.0981 


60 


9.4260 


9-3635 


9-4374 


9-3369 


9^4515 


9.3010 


9.4685 


9-2530 


9.4884 


9.1874 


9-5115 


9.0943 



a = A X AS X tan «|> b=BxA8tan8 At = a + b 

Middle Chronometer time -f At = Chronometer time of apparent noon. 



r 



T-A-EXj-B 2 



Values of Log A and Log B for *' Equal Altitudes. 



For noon 
A- B- 


- 


ARGUMENT = 


= ELAPSED TIME 


• 


For 
A 


midnight, 
4- B- 


m 


l6h 


lyh 


i8h 


igh 


20 h 


2|h 




LogAjLogB 

9-7895,9-4884 


Log A Log B 


Log A 


LogB 


Log A 


LogB 


Log AjLog B 


Log A 


LogB 


o 


9-85399-6383 


9-9287 


9-7782 


0.0172 


9.9167 


0.1249 


0.0625 


0,2623 


0.2279 


2 


•7915 


-4937 


.8562i .6431 


-9314 


-7827 


.0204 


.9213 


.1290 


.0676 


.2676 


•2339 


4 


•7935 


.4990 


.8585 .6478 


-9341 


-7873 


.0237 


.9260 


•1330 


.0727 


.2729 


.2401 


6 


•7955 


.5042 


.8608' 


-6526 


•9368 


.7919 


.0270 


.9307 


-1371 


.0779 


-2783 


.2462 


8 


•7975 


-5094 


-8632 


-6573 


•9396 


•7965 


.0303 


.9355 


.1412 


.0830 


-2838 


•2524 


lO 


9.7996 


9-5146 


9-8655 


9.6621 


9-9424 


9.8011 


0.0336 


9.9402 


0. 1454 


0.0882 


0,2893 


0.2587 


12 


.8016 


•5197 


.8679 


.6668 


-9451 


-8057 


.0370 


• 9449 


.1496 


-0935 


-2949 


.2650 


'4 


•8037 


-5248 


-8703 


-6715 


-9479 


-8103 


.0403 


-9497 


.1538 


.0987 


-3005 


.2714 


i6 


•8058 


-5300 


•8727 


.6762 


.9508 


.8149 


-0437 


.9544 


.1581 


.1040 


•3063 


.2778 


i8 


.8078 


-5351 


-8751 


.6809 


-9536 


-8195 


.0472 


•9592 


.1623 


.1093 


.3120 


.2843 


20 


9.8099 


9-5401 


9-8775 


9-6856 


9-9564 


9.8241 


0,0506 


9.9640 


0.1667 


0. 1 146 


0.3179 


0.2909 


22 


.8120 


-5452 


.8799 


-6903 


•9593 


.8287 


.0541 


.9687 


.1711 


.1200 


-3238 


-2975 


24 


.8141 


-5502 


.8824 


-6949 


.9622 


-8333 


.0576 


-9735 


•1755 


-1253 


-3298 


.3041 


26 


.8162 


-5553 


.8848 


.6996 


-9651 


-8379 


,0611 


-9784 


.1799 


.1308 


-3359 


.3109 


28 


.8184 


-5603 


-8873 


-7043 


.9680 


.8425 


.0646 


-9832 


.1844 


.1362 


.3420 


-3177 


30 


9-8205 


9-5653 


9.8898 


9.7089 


9.9709 


9.8471 


0.0682 


9.9880 


0.1889 


0.1417 


0.3482 


0.3245 


32 


.8227 


.5702 


-8923 


.7136 


•9739 


.8^:17 


.0718 


-9929 


.1935 


.1472 


-3545 


•3315 


34 


.8248 


-5752 


.8948 


.7182 


-9769 


-8563 


-0754 


-9977 


.1981 


-1527 


.3609 


-3385 


36 


.8270 


-5801 


•8973 


.7228 


.9798 


.8609 


.0790 


0.0026 


.2028 


.15S2 


-3674 


.3456 


^ 


.8292 


.5850 


-8999 


-7275 


.9829 


.8655 


.0827 


.0075 


•2075 


.1638 


•3739 


.3527 


40 


9-8314 


9.5900 


9.9024 


9-7321 


9-9859 


9.8701 


0.0864 


0.0124 


0.2122 


0.1695 


0.3805 


0-3599 


42 


-8336 


-5948 


.9050 


-7367 


.9889 


.8748 


.0901 


.0173 


.2170 


-1751 


.3873 


-3673 


44 


-8358 


•5997 


•9075 


-7413 


.9920 


-8794 


-0939 


.0223 


.2218 


.1808 


-3941 


•3747 


46 


-8380 


.6046 


.9101 


.7459 


-9951 


.8840 


.0976 


.0272 


.2267 


.1866 


.4010 


.3822 


48 


.8402 


.6094 


.9127 


.7505 


.9982 


.8887 


.1015 


.0322 


.2316 


.1924 


.4080 


-3897 


50 


9-8425 


9-6143 


9-9154 


9-7552 


0.0013 


9-8933 


0- 1053 


0.0372 


0.2366 


0.19S2 


0.4151 


0-3974 


52 


-8447 


.6191 


.9180 


•7598 


.0044 


.8980 


.1092 


.0422 


.2416 


.2040 


.4223 


.4052 


54 


.8470 


.6239 


.9206 


.7644 


.0076 


.9026 


.1131 


•0473 


.2467 


.2099 


.4297 


.4130 


56 


-8493 


.6287 


-9233 


.7690 


.0108 


-9073 


.1170 


-0523 


.2518 


■2159 


•4371 


.4210 


58 


.8516 


.6335 


.9260 


•7736 


.0140 


.9120 


.1209 


-0574 


.2570 


.2219 


.4446 


.4291 


60 


9-8539 


9-6383 


9-9287 


9.7782 


0.0172 


9.9167 


0.1249 


0.0625 


0.2623 


0,2279 


0.4523 


0.4372 



a = A X AS X tan <|) b = B X A8 tan 8 At = a + b 

Middle Chronometer time -f At = Chronometer time of apparent noon. 



Solar ParaMax. 



Altitude. 


Solar Pa 


rallax in A 


iltitade. 


h'-\-r-\-s 


8.70 


8.80 


8.9c 


io° 


8.57 


8.67 


8.77 


• 5 


8.40 


8.50 


8.60 


20 


8.18 


8.27 


8.36 


25 


7.89 


7.98 


8.07 


28 


7.68 


1-11 


7.86 


30 


7-53 


7.62 


7.71 


32 


7.38 


7.46 


7-55 


34 


7.21 


7.30 


7.38 


36 


7.04 


7.12 


7.20 


38 


6.86 


6.93 


7.01 


40 


6.67 


6.74 


6.82 


42 


6.47 


6.54 


6.61 


44 


6.26 


6.33 


6.40 


46 


6.04 


6.11 


6.18 


48 


5.82 


5.89 


5.96 


50 


5.59 


5-66 


5-72 


52 


5.36 


5-42 


5-50 


54 


5.11 


5-17 


5-23 


56 


4.87 


4.92 


4.98 


58 


4.61 


4.66 


4.72 


60 


4.35 


4.40 


4.45 


62 


4.08 


4.13 


4.18 


64 


3-81 


3.86 


3-90 


66 


3-54 


3.58 


3.62 


68 


3.26 


3-30 


3-33 


70 


2.98 


3.01 


3-04 


72 


2.69 


2.72 


2.75 


74 


2.40 


2.43 


2.45 


76 


2.10 


2.13 


2.15 


78 


1. 81 


1.83 


1.85 


80 


I-5I 


1-53 


1-55 



Horizontal Parallax = ir Am. Eph. 278 
Parallax in Altitude = ir cos {/i^-{-r-\~ ?) 



Log K for "Circum meridians. 



DAIlv 


HOURLY 
RATE. 


Values of Log ^. 


RATE. 


IFGAINING 


IF LOSING 


O^ 


0^000 


0.000 0000 


0.000 0000 


2 


.083 


9.999 9799 


0201 


4 


.167 


9598 


0402 


6 


.250 


9397 


0603 


8 


-333 


9196 


0804 


lO 


.417 


9.999 8995 


0.000 1005 


12 


.500 


8794 


1206 


14 


.583 


8593 


1407 


i6 


.667 


8392 


1608 


i8 


.750 


8191 


1809 


20 


.833 


9.999 7990 


0.000 2010 


22 


.917 


7789 


22II 


24 


<.ooo 


7588 


2412 


26 


.083 


7387 


2613 


28 


.167 


7186 


2814 


30 


<.250 


9.999 6985 


0.000 3015 


32 


•333 


6784 


3216 


34 


.417 


6583 


3417 


36 


.500 


6382 


3618 


38 


.583 


6181 


3819 


40 


1.667 


9.999 5980 


0.000 4020 


42 


.750 


5779 


4221 


44 


.833 


5578 


4422 


46 


.917 


5377 


4623 


48 


2.000 


5176 


4824 


50 


2.083 


9-999 4975 


0.000 5025 


52 


.167 


4774 


5226 


S4 


.250 


4573 


5427 


56 


-333 


4372 


5628 


58 


-4«7 


4171 


5829 


60 


2.500 


9-999 3970 


0.000 6030 



r 3600 1 

[3600- Rj 



f 86400 ] 

[86400 -rJ 



T-^EXjE 5 



Values ofm for ** Reduction to the Meridian. 



">i 



s 


om 


,m 


2m 


3m 


4m 


5m 


5m 


ym 


8m 


Qm 


o 


c/^oo 


i^^96 


7^^85 


17^^67 


31^^42 


49^-^09 


70^^68 


96^^20 


125^^65 


159^-^02 


2 


0.00 


2.10 


8.12 


18.37 


31-94 


49-74 


71.47 


97.12 


126.70 


160.20 


4 


O.OI 


2.23 


8.39 


18.47 


32.47 


50.40 


72.26 


98.04 


127-75 


161.39 


6 


0.02 


2.38 


8.66 


18.87 


33-OI 


51.07 


73.06 


98-97 


128.81 


162.58 


8 


0.03 


2.52 


8.94 


19.28 


33-54 


51.74 


73.86 


99.90 


129.87 


163.77 


lO 


0.05 


2.67 


9.22 


19.69 


34-09 


52.41 


74.66 


100.84 


130.94 


164.97 


12 


0.08 


2.83 


9.50 


20.11 


34-64 


53.09 


75-47 


101.78 


132.01 


166.17 


•4 


0. II 


2.99 


9-79 


20.53 


35-19 


53.77 


76.29 


102.72 


133.09 


167.37 


i6 


0.14 


3-15 


10.09 


20.95 


35-74 


54.46 


77.10 


103.67 


134.17 


168.58 


iS 


0.18 


3-32 


10.39 


21.38 


36-30 


55.15 


77-93 


104.63 


135.25 


169.80 


20 


0.22 


3-49 


10.69 


21.82 


36.87 


55.84 


78.75 


105.53 


136.34 


171.02 


22 


0.26 


3.67 


11.00 


22.25 


37-44 


56.55 


79-58 


106.55 


137.43 


172.24 


24 


0.31 


3.85 


II. 31 


22.70 


38.01 


57-25 


80.42 


107.51 


138.53 


173.47 


26 


0-37 


4.03 


11.63 


23.14 


38.59 


57.96 


81.26 


108.48 


139.63 


174.70 


28 


0.43 


4.22 


11-95 


23.60 


39.17 


58.68 


82.10 


109.46 


140.74 


175.94 


30 


0.49 


4.42 


12.27 


24.05 


39.76 


59.39 


82.95 


110.44 


141.85 


177.18 


32 


0.56 


4.62 


12.60 


24.51 


40.35 


60.11 


83.81 


III. 43 


142.96 


178.43 


34 


0.63 


4.82 


12.93 


24.98 


40.95 


60.84 


84.66 


112. 41 


144.08 


179.68 


36 


0.71 


5.03 


13.27 


25-45 


41.55 


61.57 


85.52 


113.40 


145.20 


180.93 


38 


0.79 


5-24 


13.62 


25.92 


42.15 


62.31 


86.39 


114.40 


146.33 


182.19 


40 


0.87 


5-45 


13.96 


26.40 


42.76 


63.05 


87.26 


115.40 


147.46 


183.46 


42 


0.96 


5.67 


14-31 


26.88 


43-37 


63.79 


88.14 


116.40 


148.60 


184.72 


44 


1.06 


5-90 


14.67 


27.37 


43-99 


64.54 


89.01 


117.41 


149.74 


185.99 


46 


I-I5 


6.13 


15.03 


27.86 


44.61 


65.29 


89.89 


118.43 


150.88 


187.27 


48 


1.26 


6.36 


15.39 


28.35 


45-24 


66.05 


90.78 


119-45 


152.03 


188.55 


50 


1.36 


6.60 


15-76 


28.85 


45-87 


66.81 


91.68 


120.47 


153.19 


189.83 


52 


1.48 


6.84 


16.14 


29.36 


46.50 


67.58 


92.57 


121.49 


154.35 


191. 12 


54 


1-59 


7.09 


16.51 


29.86 


47.14 


68.35 


93.47 


122.53 


155.51 


192.41 


56 


1. 71 


7-34 


16.89 


30.38 


47.79 


69.12 


94.38 


123-57 


156.67 


193.71 


58 


1.83 


7.60 


17.28 


30.90 


48.43 


69.90 


95-29 


124.61 


157.84 


195.01 


60 


1.96 


7.85 


17.67 


31.42 


49.09 


70.68 


96.20 


125-65 


159.02 


196.32 




Ar 


gumer 


lt==:H0 


ur ang 


e, t; 




in = 3 


.sin'^>^ t 


cosec 1 


'/ 



r 






T J^BT ."Fl 5 








Values of m for "Reduction to the Meridian." 




s 


,om 


II m 


12"! 


13m 


14m 


15m 


16^" 


lym 


18^" 


o 


196^^32 


237''54 


282'''68 


33i''74 


384^^74 


441^^63 


502^^46 


567^^2 


635^^3 


2 


197.63 


238.98 


284.26 


333-44 


386.56 


443-60 


504-55 


569-4 


638.2 


4 


198.94 


240.42 


285.83 


335-15 


388.40 


445-56 


506.65 


571-6 


640.6 


6 


200.26 


241.87 


287.41 


336.86 


390.24 


447-54 


508.76 


573-9 


642.9 


8 


201.59 


243-33 


289.00 


338.58 


392.09 


449-51 


510.86 


576.1 


645-3 


lO 


202.92 


244.79 


290.58 


340.30 


393.94 


451-50 


512.98 


578.4 


647.7 


12 


204.25 


246.25 


292.18 


342 02 


395.79 


453-48 


515-09 


580.6 


650.0 


«4 


205.59 


247.72 


293-78 


343-75 


397-65 


455-47 


517.21 


582.9 


652.4 


i6 


206.93 


249.19 


295-38 


345-49 


399-52 


457-47 


519-34 


585.1 


654-8 


i8 


208.27 


250.67 


296.99 


347-23 


401.38 


455-47 


521.47 


587-4 


657.2 


20 


209.62 


252.15 


298.60 


348.97 


403-26 


461.47 


523-60 


589.6 


659-6 


22 


210.98 


253-63 


300.21 


350.71 


405.14 


463-48 


525-74 


59^-9 


662.0 


24 


212.34 


255.12 


301-83 


352.46 


407.02 


465-49 


527-89 


594-2 


664.4 


26 


213.70 


256.62 


303.46 


354-22 


408. 90 


467-51 


530.03 


596.5 


666.8 


28 


215.07 


258.12 


305-09 


355-98 


410.79 


469-53 


532.18 


598.7 


669.2 


33 


216.44 


259.62 


306.72 


357-74 


412.68 


471-55 


534-33 


601.0 


671.6 


32 


217.81 


261.12 


308.36 


359-51 


414-59 


473-58 


536.50 


603.3 


674-1 


34 


219.19 


262.64 


310.00 


361.28 


416.49 


475-62 


538.67 


605.6 


676.5 


35 


220.58 


264.15 


311-65 


363-07 


418.40 


477-65 


540.83 


607.9 


678.9 


38 


221.97 


265.68 


313-30 


364-85 


420.31 


479.70 


543-00 


610.2 


681.4 


43 


223.36 


267.20 


3M-95 


366.64 


422.23 


481.74 


545-18 


612.5 


68 -V 8 


42 


224.76 


268.73 


316.61 


368.42 


424.15 


483.79 


547-36 


614.8 


686.2 


44 


226.16 


270.26 


318.27 


370.21 


426.07 


485.85 


549-55 


617.2 


688.7 


46 


227.57 


271.79 


319-94 


372.01 


428.01 


487.91 


551-73 


619-5 


691. 1 


43 


228.98 


273-34 


321.62 


373-82 


42993 


489.97 


553-93 


621.8 


693.6 


50 


230.39 


274.88 


323.29 


375-62 


431-87 


492.05 


556.13 


624.1 


696.0 


52 


231.81 


276.43 


324-97 


377-43 


433-82 


494.12 


558.34 


626.5 


698.5 


54 


233-24 


277.98 


326.66 


379-26 


435-76 


496.19 


560.55 


628.8 


701.0 


56 


234.67 


279-55 


328.35 


381.08 


437-71 


498.28 


562.76 


631.2 


703-5 


58 


236.10 


281.12 


330.04 


382.90 


439-67 


500.37 


564-98 


633-5 


705-9 


60 


237.54 


282.68 


331.74 


384.74 


441.63 


502.46 


567-19 


635-8 


708.4 


L 


Argu 


ment = 


Hour a 


ngle, t; 




t)t = 2 


sin2 i^t 


cosec 1^^ 





TJ^^BXjE 5 



Values of m and n for " Reduction to the Meridian." 



s 


ag"! 


20m 


21"! 


22"i 


23 m 


24m 


t 


n 




t 


n 


o 


708^^4 


784^^9 


855^:^ 


949/^6 


1037^^8 


1 1 29^-^9 


,m o-^ 


0^^000 


•3 


m qS 


0^^267 


2 


710.9 


787-5 


868.0 


952.4 


1040.8 


1 133-0 


20 


.000 




20 


0.296 


4. 


713-4 


790.1 


870.8 


955-3 


1043.8 


1136.2 


40 


.000 




40 


0.327 


6 


715-9 


792.7 


873-5 


958.2 


1046.8 


1139-3 


2 


.000 


«4 





0-359 


a. 


718.4 


795-4 


876.3 


961. 1 


1049.8 


1142.5 


20 
40 


.000 
.001 




20 
40 


0.394 
0.432 


lO 


720.9 


798-0 


879.0 


963-9 


1052.8 


1145.6 


3 


.001 


'5 





0.473 


12 


723-4 


800.7 


881.8 


966.9 


1055-9 


1148.8 


20 


.002 




20 


0.517 


"4 


725-9 


803.3 


884.6 


969.8 


1058.9 


1152.0 


40 


.002 




40 


0.563 


i6 


728.4 


806.0 


8S7.4 


972.7 


1062.0 


1155-2 


4 


.002 


16 





0.612 


i8 


730-9 


808.6 


890.2 


975-5 


1065.0 


1158.3 


20 

43 


.003 
.005 




20 
40 


0.665 
0.721 


2D 


733-5 


811. 3 


893.0 


978.5 


1068. 1 


1161.5 


5 


.007 


•7 





0.780 


22 


736.0 


813-9 


895-8 


981.4 


1071.1 


1 1 64. 7 


20 


.009 




20 


0.843 


24 


738-5 


816.6 


89S.6 


984.4 


1074.2 


1167.9 


40 


.011 




40 


0.910 


26 


741. 1 


8*19.2 


901.4 


987-3 


1077.2 


1171.1 


6 


.012 


18 





0.980 


28 


743-6 


821.9 


904.2 


990.3 


1080.3 


1^74-3 


20 
40 


.015 
.018 




20 
40 


1.054 
1-133 


30 


746.2 


824.6 


907.0 


993-2 


1083.3 


1177-5 


7 


.022 


19 





1. 216 


32 


748.7 


827.3 


909.8 


995.2 


1086.4 


1 180.7 


20 


.026 




20 


1.304 


34 


751-3 


829.9 


912.6 


999.1 


1089.5 


1183.9 


40 


-032 




40 


1-397 


36 


753-8 


832.6 


915.5 


1002. 1 


1092.6 


1187.1 


8 


.038 


20 





1-493 


38 


756.4 


835.3 


918.3 


1005.0 


1095-7 


1 190.3 


20 
40 


-045 
-053 




20 
40 


1-595 
1.702 


40 


759-0 


838.0 


921. 1 


1008.0 


1098.8 


1 193- 5 


9 


.061 


2i 





1. 815 


42 


761.5 


840.7 


923-9 


1010.9 


1101.9 


1 196. 7 


20 


.071 




20 


1-933 


44 


764.1 


843-4 


926.8 


1013.9 


1 105.0 


1199.9 


40 


.081 




40 


2.057 


46 


766.7 


846.1 


929.6 


1016.9 


1 108. 1 


1203. 1 


10 


•093 


22 





2.186 


48 


769.3 


848.9 


932.4 


1019.9 


nil. 2 


1206.4 


20 
40 


.106 
.121 




20 
40 


2.321 
2.463 


50 


771.9 


851-6 


935-2 


1022.8 


1114.3 


1209.6 


1 1 


• 137 


23 





2. 611 


52 


774-5 


854-3 


938.1 


1025.8 


1117.4 


1212.9 


20 


-155 




20 


2.766 


54 


777.1 


857-1 


940.9 


1028.8 


1120.5 


1216.1 


40 


.174 




40 


2.926 


56 


779-7 


859.8 


943-8 


1031.8 


1123.6 


1219.4 


12 


.194 


24 





3-094 


58 


782.3 


862.5 


946.6 


1034.8 


1126.7 


1222.6 


20 
40 


.217 
.241 




20 
40 


3.270 
3-453 


60 


784-9 


865.3 


949-6 


1037.8 


1129.9 


1225.9 


13 


0.267 


25 





3-643 



Argumcnt= Hour angle, t ; m ^ 2stn'^ )4^ cosec 1^^; n = 2 sin* }i t cosec f 



